Though, we were motivated by Adleman in [1]}, by matching half the length of DNA strands, the sharing length of suffix and prefix among the vertices made a difference in Rauzy graph and HRR (which is shown in [2]}). Now, we are interested in the question what if, we match an arbitrary length \(1\le \ell \le k-1\)  of suffix/prefix word among the vertices in a graph to form an arc? On answering this question, a new graph \(\ell -\) Rauzy graph is defined as follows and its properties are studied.
